Non-decreasing martingale couplings

For many examples of couples $(\mu,\nu)$ of probability measures on the real line in the convex order, we observe numerically that the Hobson and Neuberger martingale coupling, which maximizes for $\rho=1$ the integral of $|y-x|^\rho$ with respect to any martingale coupling between $\mu$ and $\nu$, is still a maximizer for $\rho\in(0,2)$ and a minimizer for $\rho>2$. We investigate the theoretical validity of this numerical observation and give rather restrictive sufficient conditions for the property to hold. We also exhibit couples $(\mu,\nu)$ such that it does not hold. The support of the Hobson and Neuberger coupling is known to satisfy some monotonicity property which we call non-decreasing. We check that the non-decreasing property is preserved for maximizers when $\rho\in(0,1]$. In general, there exist distinct non-decreasing martingale couplings, and we find some decomposition of $\nu$ which is in one-to-one correspondence with martingale couplings non-decreasing in a generalized sense.